arXiv · 0705.3366
We construct all planar semimodular lattices in three simple steps from the direct product of two chains.
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G. Grätzer, E. Knapp. 2007-05-22. Notes on planar semimodular lattices. I. Construction. https://arxiv.org/abs/0705.3366
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A connection and a nonzero parallel adjoint field determine an invariant hyperplane constraint on a principal bundle. Its sign mirror preserves the hyperplane and reverses its coorientation; global gauge realization is controlled by a twisted stabilizer reduction. For regular fields we identify the normalizing gauge extension as a pushout of the torus-normalizer extension, giving exact lift orders and simultaneous-splitting criteria. In singular rank-two block families, reductions on a fixed trivial bundle form an affine second-Chern lattice whose Weyl stabilizers and finite-order lift spectra detect topology invisible to paired curvature. The reduction framework also determines the structure group and second cohomology of the matched-flag diagonalization space of Friedman and Park, and gives a first- and second-Chern criterion for normal matrices with fixed separated spectrum on four-complexes; every integral solution of their three-eigenline equation on $S^2\times S^2$ is realized. For moving reductions, the projected circle curvature differs from the ambient paired curvature by a covariant-derivative term. Full fatness on a closed four-manifold forces a nontrivial sign-mirror obstruction for every circle reduction; hyperbolic self-dual-form bundles also provide circle reductions in the $y$-fat setting of Florit and Ziller. Contact transgression, bundle automorphism twists, and the natural first-jet Spencer operator complete the geometric picture.
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